In a large pot, melt 2 tablespoons of unsalted butter or heat olive oil over medium heat. Add 1 diced medium onion and 2 minced garlic cloves to the pot, cooking for about 3–4 minutes until fragrant and translucent.

Next, stir in 2 diced medium carrots and 2 diced medium potatoes. Sauté the mixture for about 5 minutes, stirring occasionally, until the vegetables begin to soften.

Sprinkle 1/4 cup of all-purpose flour over the sautéed vegetables, stirring until everything is evenly coated. Cook this mixture for 1-2 minutes to create a roux.

Slowly whisk in 4 cups of vegetable broth, ensuring there are no lumps. Stir until the mixture is smooth and begins to thicken, which should take about 2–3 minutes.

Sprinkle in 1 teaspoon of dried thyme, 1/2 teaspoon of dried rosemary, and season with salt and pepper to taste. Allow the soup to simmer for 15 minutes, uncovered.

Mix in 2 cups of broccoli florets, 1 cup of frozen peas, and 1 cup of corn kernels, allowing the soup to simmer for an additional 5–7 minutes.

Pour in 1 cup of whole milk or half-and-half, stirring until everything is thick and creamy. If desired, add 1/2 cup of heavy cream.

Once thickened, taste and adjust the seasoning with more salt or pepper if needed. Ladle the soup into bowls, garnishing with fresh parsley.